Permutations And Combinations
MODULE - I
very tedious. Here the Fundamental Principle of Counting or simply the Counting Principle
comes in use :
If any event can occur in m ways and after it happens in any one of these ways, a
second event can occur in n ways, then both the events together can occur in m× n
Example 7.1 How many multiples of 5 are there from 10 to 95 ?
Solution : As you know, multiples of 5 are integers having 0 or 5 in the digit to the extreme right
(i.e. the unit’s place).
The first digit from the right can be chosen in 2 ways.
The second digit can be any one of 1,2,3,4,5,6,7,8,9.
i.e. There are 9 choices for the second digit.
Thus, there are 2 × 9 = 18 multiples of 5 from 10 to 95.
Example 7.2 In a city, the bus route numbers consist of a natural number less than 100,
followed by one of the letters A,B,C,D,E and F. How many different bus routes are possible?
Solution : The number can be any one of the natural numbers from 1 to 99.
There are 99 choices for the number.
The letter can be chosen in 6 ways.
∴ Number of possible bus routes are 99 × 6 = 594 .
CHECK YOUR PROGRESS 7.1
(a) How many 3 digit numbers are multiples of 5?
(b) A coin is tossed thrice. How many possible outcomes are there?
(c) If you have 3 shirts and 4 pairs of trousers and any shirt can be worn with any pair of
trousers, in how many ways can you wear your shirts and pairs of trousers?
(d) A tourist wants to go to another country by ship and return by air. She has a choice of
5 different ships to go by and 4 airlines to return by. In how many ways can she perform
(a) In how many ways can two vacancies be filled from among 4 men and 12 women if
one vacancy is filled by a man and the other by a woman?
(b) Flooring and painting of the walls of a room needs to be done. The flooring can be
done in 3 colours and painting of walls can be done in 12 colours. If any colour combination
is allowed, find the number of ways of flooring and painting the walls of the room.
So far, we have applied the counting principle for two events. But it can be extended to
three or more, as you can see from the following examples :